Spatially distant sources of neodymium (Nd) to the ocean that carry different isotopic signatures (

Rare earth elements (REEs) have long been recognized to provide unique insight into ocean circulation and biogeochemical cycles

Neodymium is part of a long-lived isotope system. Samarium-147 (

Early measurements of

More recently, the GEOTRACES program was created to better understand the sources and cycling of trace elements and isotopes in the ocean and how they impact broader marine biogeochemical cycles. The GEOTRACES Science Plan

To gain the most useful and accurate information from these observations, however, it is paramount to understand their modern ocean biogeochemical cycles and tracer budgets. Neodymium and other REEs enter the ocean via rivers, submarine groundwater discharge, eolian deposition, pore waters, and/or interaction with sediments (Fig.

Models of the marine Nd cycle, in conjunction with seawater measurements, offer a way to constrain the magnitudes and isotopic compositions of these various inputs to the ocean and identify the most important sources. Of the distinct types of models, the following four have been used to simulate the modern ocean Nd cycle: simple box models, ocean general circulation models (OGCMs), steady-state circulation models, and boundary propagation models, each with their strengths and weaknesses (Table

Previous modeling studies. Note: GCM is general circulation model, and TMI is total matrix intercomparison.

Box models typically refer to models consisting of 10 or fewer well-mixed boxes that exchange tracer with each other through prescribed mixing and overturning rates. Owing to their small size, box model simulations are the fastest to run and require very little computational power. Thus, they facilitate parameter optimization and scientific exploration by allowing for quick experimentation. For example, box models have been successfully used to determine that Nd must exchange between seawater and particles in the water column or at the sediment–water interface

Ocean general circulation models sit on the other end of the spectrum of computational complexity, with better spatial resolution and resolved physics. Their computational costs generally prohibit systematic parameter space exploration or parameter optimization. These models have thus been used primarily to run well-defined experiments that target specific hypotheses, such as the importance of continental margin sources (boundary exchange) on

More recently, a new class of steady-state models has emerged with unique potential to combine the advantages of OGCMs with the computational speed of box models. These models do not resolve the physics at runtime and, instead, rely on a prescribed, steady-state circulation. They can thus directly solve for the steady-state solution of the three-dimensional tracer equations, avoiding costly spin-ups, and drastically reducing simulation times. Thus far, to our knowledge, these models have only been used to test the top-down hypothesis by propagating a surface boundary condition into the ocean interior. Using the transport matrix method

The fourth class of models, which we have termed boundary propagation models, entirely bypasses expressing fluxes between model grid cells by connecting interior grid cells directly to the surface, using the total matrix intercomparison method

Our goal is to fill the current gap in the marine Nd modeling landscape and leverage the largely unexplored benefits of steady-state circulation models. Hence, here, we present the Global Neodymium Ocean Model (GNOM) v1.0, a mechanistic model of the modern ocean Nd cycle embedded in a state-of-the-art, steady-state estimate of the modern ocean circulation from the Ocean Circulation Inverse Model version 2

The GNOM v1.0 thus provides the community with a realistic yet computationally affordable tool to model the marine Nd cycle that we hope will be used to further improve our understanding of Nd cycling in the ocean. The model code and its optimization script are available publicly on GitHub at

Additionally, the steady-state formulation of the GNOM is amenable to novel Green-function-based diagnostics that can provide important new insights into major open questions on the marine Nd cycle. Green functions (sometimes spelled Green's functions) can be used for solving ordinary differential equations with an initial condition and/or boundary values

Neodymium concentrations are controlled by the interplay between circulation, external sources, and reversible scavenging and burial in the sediments (Fig.

Diagram of the Nd cycle model as implemented in GNOM v1.0. External sources of dissolved Nd are represented by black arrows. Localized sources, rivers, groundwater, and hydrothermal vents are indicated by a small circle at the origin of their respective arrows. A fraction of Nd is reversibly scavenged and pumped downwards. A fraction of scavenged Nd that reaches the sediments is buried in the sediments and removed from the system. Nd is also continuously transported by the ocean circulation model (not represented in the schematic).

The global

The

The GNOM v1.0 explicitly represents the following six sources of Nd into the ocean (Fig.

Vertically integrated Nd sources and corresponding vertical mean

We assume that atmospheric dust deposition injects

The isotopic signature of atmospheric mineral dust deposited on the ocean surface is not homogeneous

Extent of the dust regions of origin

We assign a distinct Nd solubility and isotopic signature to each region of origin, controlled by the

Each isotopic signature parameter

Despite a smaller atmospheric loading than mineral dust, we include volcanic ash as a separate, potentially important, eolian source of Nd because of its typically high reactivity and solubility compared to mineral dust. This reactivity partly reflects the high surface area of volcanic ash and the thermodynamic instability of volcanic glass

Sedimentary Nd is likely released via pore waters located in the upper few centimeters below the seafloor

There is no established quantitative flux model for sedimentary Nd release that works on the global scale, especially given the limited spatial coverage of direct sedimentary flux measurements (which are almost entirely restricted to the coastal northwestern Pacific). Therefore, the GNOM v1.0 implements the sedimentary Nd flux into the ocean as a flexible and optimizable function of depth

The base sedimentary flux is further scaled by a reactivity factor,

Finally, to account for large glaciers that may produce fine-grained glacial flour from previously unexposed bedrock that likely contains reactive Nd

For the

For riverine sources, we use the Global River Flow and Continental Discharge Dataset

Riverine

Neodymium also enters the oceans via coastal groundwater

Following

A minor fraction of the marine neodymium budget presumably comes from hydrothermal vents, which deliver likely young Nd (high

Arguably, the hydrothermal system as a whole acts as a net sink of Nd in the ocean

Neodymium is removed from the system through scavenging onto particles. We follow, e.g.,

However, we avoid the explicit simulation of scavenged neodymium,

We consider four different particle types for scavenging Nd.
(i) Scavenging by dust particles is modeled using the dust deposition fields of

This fourth scavenging mechanism effectively behaves like spontaneous precipitation, and, as such, will be referred to as precipitation throughout this study (subscript prec). Precipitation is implemented by using a spatially uniform fictitious particle concentration of

The output of our model is governed by a set of

The mismatch with each observation is quantified by the square of the difference between the observed value and the modeled value from the closest grid cell. Because we use observations of

In Eq. (

The third term of Eq. (

For the performance and robustness of the optimization, we additionally perform a variable transform

The

The objective function depends on

The

Locations of

We use the Newton trust region algorithm from the Optim.jl package

Thanks to the computationally efficient gradient optimization algorithm that leverages gradient and Hessian information, the entire optimization run takes a few hours on a modern laptop. In our experience, for comparison, using the more standard finite differences approach or an optimization algorithm that does not have access to derivatives would likely take multiple months.

Because the Newton trust region algorithm performs local rather than global optimization, we run multiple optimizations, starting from randomized initial parameter values sampled from the parameter distributions

Our best estimate of the state of the Nd cycle is given by the set of parameters that minimizes the objective function defined in Eqs. (

List of parameters. Realistic parameter ranges were prescribed based on the literature and the expertise of the authors. Final values have been rounded to three significant digits. Parameters without a range are not optimized (final value equals initial guess). Scavenging reaction constants,

In Table

The general fit to observations is illustrated in Fig.

Quantiles of the cumulative joint probability density functions of modeled and observed

While statistics such as Fig.

We explore the regional variations in the model's skill with depth in Fig.

Basin-averaged profiles of

We further assess the model skill by looking at GEOTRACES transects individually. Out of the 3483 observations of [Nd] that we use to constrain our model, 1575 (

Figure

Figure

Figure

Model (heatmaps) and observations (markers) for

One of the biggest advances in the GNOM v1.0, compared to earlier models of the marine Nd cycle, is due to the steady-state matrix formulation of the model, which allows us to compute advanced and detailed diagnostics that can directly address fundamental questions about the distribution of tracers and better understand their cycle. In the following sections, we showcase a few such diagnostics.

The optimized parameters determine the magnitude of the sources, which are collected in Table

The eolian dust and sedimentary sources are the dominant ones contributing

At

Source magnitudes, their Nd contributions, and corresponding bulk residence time. Values are reported with two significant digits.

We partition Nd concentration according to source type simply by removing all the other sources.

As is the case for most global biogeochemical cycles, the relative source magnitudes and their relative contribution to the standing stock do not necessarily match. For instance, mineral dust, volcanic ash, and hydrothermal vents contribute more to the mean Nd concentration than their relative source magnitudes. These variations can be directly linked to the bulk residence time of Nd molecules, which vary with location of injection and consequently with source type.

The bulk residence time of

Unlike in the real ocean, where each molecule of Nd is indistinguishable from the next (with no information about its initial source), in a model we can track Nd coming from different sources and calculate source-partitioned residence times. We find that sedimentary-sourced Nd has the shortest residence time at

Sediment core records of

Our GNOM model – or, more precisely, the steady-state matrix schema with which it is built – allows for exact computations of these water masses and the contributions of various sources and regions to the modern ocean Nd and

We chose two simple regions that cover the entire ocean, except for the central Atlantic between

Northern Atlantic (

First, we track Nd concentrations from each of these regions.
Neodymium concentrations are not conservative, in part due to reversible scavenging and in part due to external sources that inject new Nd along transport pathways. For example, we can track Nd that came from region

Figure

We now track

Water mass fractions have been estimated using a Green function boundary propagator in similar model contexts

The most prominent caveat of GNOM v1.0 is the steady-state assumption, which we apply to both the circulation and the Nd cycle. Hence, by construction, daily, seasonal, decadal, or multi-decadal fluctuations that deviate from the climatological mean cannot be captured by our model. However, we trust that the circulation model used

We note that, compared to previous modeling studies, the GNOM does not represent scavenging by calcium carbonate (CaCO_{3}) because there is no publicly available particulate CaCO_{3} field, to the best of our knowledge. Modeling scavenging is a challenging task that the GNOM model does not pretend to achieve with high accuracy.
However, we deem the current implementation satisfactory, considering the quality of the overall model–observation fit.
Future versions of the GNOM could include CaCO_{3} particle scavenging or a generally improved scavenging parameterization.

Our model reveals some locations of particular interest for improving our understanding of the Nd cycle and

While our model endeavors to use formulations and parameter constraints which have reasonable biogeochemical interpretations, there is always room for improvement. For example, our sedimentary source parameterization, which we plan to investigate further in future work, assumes an exponential profile for the sedimentary source flux. While this parameterization is flexible enough to reproduce most of the qualitative features of the sedimentary fluxes in previous models, one might argue that a more mechanistic model of the sedimentary budget would result in a more realistic overall Nd cycling model.

Despite the theoretical advantages they confer to the convergence rate of our optimization procedure, our specific choice of prior distributions for the parameters (Fig.

Our specific choice of objective function gives a measure of the skill of the model for reproducing

Qualitatively, GNOM compares well to previous models that are embedded in ocean general circulation models and simulate two explicit Nd isotopes

The main advantages of the GNOM are skill and computational efficiency. The GNOM v1.0 owes its low computational cost and quick simulation time to the steady-state OCIM v2.0 circulation in which it is embedded and the linear representation that allows us to solve the system of tracer equations in a single matrix inversion. The model's skill comes from the optimization procedure and likely benefits from the quality of the OCIM v2.0 circulation. The GNOM is also versatile in many respects owing to its simplicity. Parameter values and acceptable ranges can be tuned, entire mechanisms can be turned off, eliminating free parameters, or added with a few changes of simple lines of code.
This versatility is compounded by computational speed, which makes GNOM ideally suited for quick experimentation and further optimization. The GNOM model is also easily diagnosed, owing to the powerful tools of linear algebra. Novel diagnostics offer new insights by revealing features often hidden in standard model output. Finally, as it is available in a self-contained package (except for the GEOTRACES dataset, which is not programmatically accessible), the GNOM v1.0 offers unprecedented reproducibility, which is sorely lacking in advanced research

To represent scavenging by opal (particulate silica), we designed and optimize a simple Si cycling model in parallel to our Nd cycling model. Our Si cycling model is a simple nutrient restoring model embedded in the same OCIM v2.0 circulation. We emphasize that the goal here is only to generate a reasonable 3D field for particulate biogenic silica concentrations.

The Si cycling model considered here explicitly tracks two tracers, DSi and PSi. We thus denote the modeled column vectors for DSi and PSi concentrations by the following:

All the silicate that is taken up in this model is converted to sinking particulate PSi, which gravitationally settles with optimizable velocity parameter

Quantiles of the cumulative joint probability density functions of modeled and observed dissolved Si concentrations for the parallel Si cycling model. Darker colors indicate high density of data, such that

Optimized Si cycling model parameters.

Hence, the steady-state tracer equation for dissolved silicate (DSi) is as follows:

A similar optimization procedure as for the Nd cycle is applied to optimize the five parameters collected in Table

Figure

Parameter prior distributions (color-filled densities) and initial and optimized parameter values for a dozen of optimization runs (lines). Each line starts by showing the initial parameter value at the top and is connected to the final optimized value at the bottom. The thicker blue line represents the optimization run that was used as our best estimate.

This appendix describes the effect that enhanced Nd release with extreme

From the moments of a normal distribution, one can show, in the following, that:

The GNOM model code is open source and publicly available for free. An archive of the GNOM v1.0.2 code used in this study is hosted permanently on Zenodo at

The code is written in Julia, which is itself free and open source

The GNOM v1 model was designed using the open-source AIBECS framework, available as a Julia package

Except for the GEOTRACES IDP17 data, which must be downloaded manually (see

the OCIM v2.0 circulations by

the two-dimensional dust deposition fields partitioned according to region of origin from

the aerosol-type partitioned dust source that includes volcanic ash as used by

the riverine discharge dataset from

the groundwater discharge from coastal sheds dataset compilation from

the hydrothermal

the particulate organic carbon three-dimensional fields from

The pre-GEOTRACES IDP17 historical dataset for Nd and

Except for the model schematic (Fig.

SKVH, BP, SLG, and SGJ designed the study. HL wrote preliminary MATLAB code. BP wrote the Julia model code, performed simulations, and analyzed the data, with input from all authors. SKVH and BP wrote the original draft, and all authors contributed to revision and editing of the paper prior to submission.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank Brian Haley, two other anonymous reviewers, and the editor, for their helpful comments which substantially improved this paper.

This work has been supported by the Simons Foundation (grant no. 426570SP to Seth G. John), the National Science Foundation (grant no. OCE-1736896 to Seth G. John and grant no. OCE-1831415 to Steven L. Goldstein and Sophia K. V. Hines), the Investment in Science Fund at WHOI and the John E. and Anne W. Sawyer Endowed Fund in Support of Scientific Staff (Sophia K. V. Hines), and the Storke Endowment of the Department of Earth and Environmental Sciences, Columbia University (Steven L. Goldstein).

This paper was edited by Andrew Yool and reviewed by Brian Haley and two anonymous referees.

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